Weak Gravitational Lensing in Fourth Order Gravity

For a general class of analytic $f(R,R_{\alpha\beta}R^{\alpha\beta},R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta})$ we discuss the gravitational lensing in the Newtonian Limit of theory. From the properties of Gauss Bonnet invariant it is successful to consider only two curvature invariants between the Ricci and Riemann tensor. Then we analyze the dynamics of photon embedded in a gravitational field of a generic $f(R,R_{\alpha\beta}R^{\alpha\beta})$-Gravity. The metric is time independent and spherically symmetric. The metric potentials are Schwarzschild-like, but there are two additional Yukawa terms linked to derivatives of $f$ with respect to two curvature invariants. Considering the case of a point-like lens, and after of a generic matter distribution of lens, we study the deflection angle and the images angular position. Though the additional Yukawa terms in the gravitational potential modifies dynamics with respect to General Relativity, the geodesic trajectory of photon is unaffected by the modification in the action by only $f(R)$. While we find different results (deflection angle smaller than one of General Relativity) only thank to introduction of a generic function of Ricci tensor square. Finally we can affirm the lensing phenomena for all $f(R)$-Gravities are equal to the ones known from General Relativity. We conclude the paper showing and comparing the deflection angle and image positions for $f(R,R_{\alpha\beta}R^{\alpha\beta})$-Gravity with respect to ones of General Relativity.Comment: 11 pages, 5 figure

Evolution method and "differential hierarchy" of colored knot polynomials

We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials on such evolution parameters is very easy to find. We apply this evolution method to study of the families of knots and links which include the cases with just two parallel and anti-parallel strands in the braid, like the ordinary twist and 2-strand torus knots/links and counter-oriented 2-strand links. When the answers were available before, they are immediately reproduced, and an essentially new example is added of the "double braid", which is a combination of parallel and anti-parallel 2-strand braids. This study helps us to reveal with the full clarity and partly investigate a mysterious hierarchical structure of the colored HOMFLY polynomials, at least, in (anti)symmetric representations, which extends the original observation for the figure-eight knot to many (presumably all) knots. We demonstrate that this structure is typically respected by the t-deformation to the superpolynomials.Comment: 31 page

Gaussian distribution of LMOV numbers

Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri-Vafa partition functions. Already the very first look at this data reveals that the LMOV numbers are randomly distributed in genus (!) and are very well parameterized by just three parameters depending on the representation, an integer and the knot. We present an accurate formulation and evidence in support of this new puzzling observation about the old puzzling quantities. It probably implies that the BPS states, counted by the LMOV numbers can actually be composites made from some still more elementary objects.Comment: 23 page